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## G = D4×D23order 368 = 24·23

### Direct product of D4 and D23

Aliases: D4×D23, C41D46, C92⋊C22, D923C2, C221D46, D462C22, C46.5C23, Dic231C22, C232(C2×D4), (C2×C46)⋊C22, (C4×D23)⋊1C2, (D4×C23)⋊2C2, C23⋊D41C2, (C22×D23)⋊2C2, C2.6(C22×D23), SmallGroup(368,31)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C46 — D4×D23
 Chief series C1 — C23 — C46 — D46 — C22×D23 — D4×D23
 Lower central C23 — C46 — D4×D23
 Upper central C1 — C2 — D4

Generators and relations for D4×D23
G = < a,b,c,d | a4=b2=c23=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 620 in 54 conjugacy classes, 25 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, D4, D4, C23, C2×D4, C23, D23, D23, C46, C46, Dic23, C92, D46, D46, D46, C2×C46, C4×D23, D92, C23⋊D4, D4×C23, C22×D23, D4×D23
Quotients: C1, C2, C22, D4, C23, C2×D4, D23, D46, C22×D23, D4×D23

Smallest permutation representation of D4×D23
On 92 points
Generators in S92
(1 47 27 70)(2 48 28 71)(3 49 29 72)(4 50 30 73)(5 51 31 74)(6 52 32 75)(7 53 33 76)(8 54 34 77)(9 55 35 78)(10 56 36 79)(11 57 37 80)(12 58 38 81)(13 59 39 82)(14 60 40 83)(15 61 41 84)(16 62 42 85)(17 63 43 86)(18 64 44 87)(19 65 45 88)(20 66 46 89)(21 67 24 90)(22 68 25 91)(23 69 26 92)
(47 70)(48 71)(49 72)(50 73)(51 74)(52 75)(53 76)(54 77)(55 78)(56 79)(57 80)(58 81)(59 82)(60 83)(61 84)(62 85)(63 86)(64 87)(65 88)(66 89)(67 90)(68 91)(69 92)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23)(24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46)(47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69)(70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92)
(1 26)(2 25)(3 24)(4 46)(5 45)(6 44)(7 43)(8 42)(9 41)(10 40)(11 39)(12 38)(13 37)(14 36)(15 35)(16 34)(17 33)(18 32)(19 31)(20 30)(21 29)(22 28)(23 27)(47 92)(48 91)(49 90)(50 89)(51 88)(52 87)(53 86)(54 85)(55 84)(56 83)(57 82)(58 81)(59 80)(60 79)(61 78)(62 77)(63 76)(64 75)(65 74)(66 73)(67 72)(68 71)(69 70)

G:=sub<Sym(92)| (1,47,27,70)(2,48,28,71)(3,49,29,72)(4,50,30,73)(5,51,31,74)(6,52,32,75)(7,53,33,76)(8,54,34,77)(9,55,35,78)(10,56,36,79)(11,57,37,80)(12,58,38,81)(13,59,39,82)(14,60,40,83)(15,61,41,84)(16,62,42,85)(17,63,43,86)(18,64,44,87)(19,65,45,88)(20,66,46,89)(21,67,24,90)(22,68,25,91)(23,69,26,92), (47,70)(48,71)(49,72)(50,73)(51,74)(52,75)(53,76)(54,77)(55,78)(56,79)(57,80)(58,81)(59,82)(60,83)(61,84)(62,85)(63,86)(64,87)(65,88)(66,89)(67,90)(68,91)(69,92), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23)(24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46)(47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69)(70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92), (1,26)(2,25)(3,24)(4,46)(5,45)(6,44)(7,43)(8,42)(9,41)(10,40)(11,39)(12,38)(13,37)(14,36)(15,35)(16,34)(17,33)(18,32)(19,31)(20,30)(21,29)(22,28)(23,27)(47,92)(48,91)(49,90)(50,89)(51,88)(52,87)(53,86)(54,85)(55,84)(56,83)(57,82)(58,81)(59,80)(60,79)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)>;

G:=Group( (1,47,27,70)(2,48,28,71)(3,49,29,72)(4,50,30,73)(5,51,31,74)(6,52,32,75)(7,53,33,76)(8,54,34,77)(9,55,35,78)(10,56,36,79)(11,57,37,80)(12,58,38,81)(13,59,39,82)(14,60,40,83)(15,61,41,84)(16,62,42,85)(17,63,43,86)(18,64,44,87)(19,65,45,88)(20,66,46,89)(21,67,24,90)(22,68,25,91)(23,69,26,92), (47,70)(48,71)(49,72)(50,73)(51,74)(52,75)(53,76)(54,77)(55,78)(56,79)(57,80)(58,81)(59,82)(60,83)(61,84)(62,85)(63,86)(64,87)(65,88)(66,89)(67,90)(68,91)(69,92), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23)(24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46)(47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69)(70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92), (1,26)(2,25)(3,24)(4,46)(5,45)(6,44)(7,43)(8,42)(9,41)(10,40)(11,39)(12,38)(13,37)(14,36)(15,35)(16,34)(17,33)(18,32)(19,31)(20,30)(21,29)(22,28)(23,27)(47,92)(48,91)(49,90)(50,89)(51,88)(52,87)(53,86)(54,85)(55,84)(56,83)(57,82)(58,81)(59,80)(60,79)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70) );

G=PermutationGroup([[(1,47,27,70),(2,48,28,71),(3,49,29,72),(4,50,30,73),(5,51,31,74),(6,52,32,75),(7,53,33,76),(8,54,34,77),(9,55,35,78),(10,56,36,79),(11,57,37,80),(12,58,38,81),(13,59,39,82),(14,60,40,83),(15,61,41,84),(16,62,42,85),(17,63,43,86),(18,64,44,87),(19,65,45,88),(20,66,46,89),(21,67,24,90),(22,68,25,91),(23,69,26,92)], [(47,70),(48,71),(49,72),(50,73),(51,74),(52,75),(53,76),(54,77),(55,78),(56,79),(57,80),(58,81),(59,82),(60,83),(61,84),(62,85),(63,86),(64,87),(65,88),(66,89),(67,90),(68,91),(69,92)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23),(24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46),(47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69),(70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92)], [(1,26),(2,25),(3,24),(4,46),(5,45),(6,44),(7,43),(8,42),(9,41),(10,40),(11,39),(12,38),(13,37),(14,36),(15,35),(16,34),(17,33),(18,32),(19,31),(20,30),(21,29),(22,28),(23,27),(47,92),(48,91),(49,90),(50,89),(51,88),(52,87),(53,86),(54,85),(55,84),(56,83),(57,82),(58,81),(59,80),(60,79),(61,78),(62,77),(63,76),(64,75),(65,74),(66,73),(67,72),(68,71),(69,70)]])

65 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 23A ··· 23K 46A ··· 46K 46L ··· 46AG 92A ··· 92K order 1 2 2 2 2 2 2 2 4 4 23 ··· 23 46 ··· 46 46 ··· 46 92 ··· 92 size 1 1 2 2 23 23 46 46 2 46 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

65 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D23 D46 D46 D4×D23 kernel D4×D23 C4×D23 D92 C23⋊D4 D4×C23 C22×D23 D23 D4 C4 C22 C1 # reps 1 1 1 2 1 2 2 11 11 22 11

Matrix representation of D4×D23 in GL4(𝔽277) generated by

 1 0 0 0 0 1 0 0 0 0 0 276 0 0 1 0
,
 276 0 0 0 0 276 0 0 0 0 1 0 0 0 0 276
,
 32 1 0 0 224 33 0 0 0 0 1 0 0 0 0 1
,
 247 227 0 0 90 30 0 0 0 0 276 0 0 0 0 276
G:=sub<GL(4,GF(277))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,276,0],[276,0,0,0,0,276,0,0,0,0,1,0,0,0,0,276],[32,224,0,0,1,33,0,0,0,0,1,0,0,0,0,1],[247,90,0,0,227,30,0,0,0,0,276,0,0,0,0,276] >;

D4×D23 in GAP, Magma, Sage, TeX

D_4\times D_{23}
% in TeX

G:=Group("D4xD23");
// GroupNames label

G:=SmallGroup(368,31);
// by ID

G=gap.SmallGroup(368,31);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-23,97,8804]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^23=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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